73.144 SemidirectPair

SemidirectPair( S )

When S is a semidirect product, this function finds a faithful permutation representation P and sets up a pairing between S and P. The example illustrates c2|Xc3congs3.

    gap> agen := ac3.generators;; pgen := pc3.generators;;
    gap> a := GroupHomomorphismByImages( pc3, ac3, pgen, agen );
    GroupHomomorphismByImages( PermAut(c3), Aut(c3), [ (1,2) ], 
    [ GroupHomomorphismByImages( c3, c3, [ (1,2,3) ], [ (1,3,2) ] ) ] )
    gap> G := SemidirectProduct( pc3, a, c3 );;
    gap> G.name := "G";;  PG := SemidirectPair( G );
    rec(
      perm := Perm(G),
      sdp := G,
      s2p := OperationHomomorphism( G, Perm(G) ),
      p2s := GroupHomomorphismByImages( Perm(G), G, [(1,2)(4,5), (3,5,4)],
        [ SemidirectProductElement( (1,2), GroupHomomorphismByImages
              ( c3, c3, [ (1,3,2) ], [ (1,2,3) ] ), () ), 
        SemidirectProductElement( (), IdentityMapping(c3), (1,2,3) ) ] ))

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